A method for assigning a finite value to certain divergent formal power series by applying the Borel transform to obtain a function with better analytic properties, analytically continuing that transform if necessary, and then applying a Laplace-type integral (Borel-Laplace inversion) to recover a summed function consistent with the original asymptotics.

Principle

Convert a factorially divergent coefficient sequence into a Borel transform with improved radius of convergence; use analytic continuation to reach an integration ray and perform a Laplace integral to produce a canonical sum when singularities are handled appropriately.

Demonstration

In many differential-equation problems, the formal asymptotic series solution has coefficients growing like n!; applying the Borel transform produces a convergent power series, analytic continuation across singularities and a Laplace integral along a chosen direction yields an actual function solving the equation and matching the asymptotic expansion.

Misapplication

Using naive Borel inversion when the Borel transform has branch points or poles on the integration ray without resolving Stokes phenomena — this can produce ambiguous or incorrect sums if lateral summation or medianization is required.

Consequence

When applicable, Borel resummation recovers an analytic function whose asymptotic expansion matches the original formal series and often restores uniqueness and analytic continuation properties, clarifying the connection between formal series and actual solutions.

Reversal

A non-resummable series resists Borel summation because its Borel transform cannot be continued or the Laplace integral diverges; such divergence signals essential singularity or severe growth preventing assignment of a canonical sum by this method.

Boundary

Requires a formal series with suitable growth (typically factorial) so the Borel transform converges, plus the possibility of analytic continuation and a convergent Laplace integral along a ray; it does not apply to arbitrary divergent series nor does it obviate issues of Stokes discontinuities.

Semantic Tension

Tension with other summation procedures (Cesàro, Abel, Euler): Borel resummation is stronger for many factorially divergent series but may disagree where analytic continuation choices introduce multiplicative ambiguities tied to Stokes data.

Synthesis

Borel resummation is the three-stage recipe—Borel transform, analytic continuation, Laplace inversion—that turns appropriate divergent asymptotic expansions into canonical analytic functions, provided singularities and integration directions are managed to respect Stokes phenomena.